Practical details about running a practice are so lacking in residency, but you are right! There is nothing preventing you from doing the above. You would, however, have to meet basic regulatory criteria.
1. You have to be licensed (looks like youre on top of that)
2. You need a DEA number (and a dangerous controlled substance or DCS number depending on the state)
3. You have to follow other basic practice criteria such as maintaining medical records of any patients seen for, usually 7 years
Of course, there are a few other issues also. One is if youve ever been uncovered for malpractice during any period of practice, you will be unable in the future to get any major carrier to cover you. So for that reason alone you pretty much have to get some malpractice. In terms of doing so however, most (but not all!) carriers will deny you coverage based on your non-completion of a residency, but you can probably find something with enough looking around. Another big issue however is that you are not just responsible for the malpractice premiums, but are also liable for tail coverage if you have any change in location or break in coverage before you start your post-residency practice-- which is very expensive! (Youll have to read up on tail coverage somewhere, but to make a long explaination short it covers prior acts from when you were actively insured by an old policy). So unless you transition directly from this pre-residency practice to post-residency practice, you are going to be stuck with a huge bill for both the insurance and tail which will probably be much more than youll make seeing patients in your basement!
As for your cash only concept, most people aint going to go for that unless you charge like $50 a visit-- which I guess wouldnt be too bad with the low overhead of operating out of your basement! I dont know though how many people would want to venture down in that basement of yours without a few more trappings of legitimacy-- ie. a receptionist, a nurse.. maybe a few more patients waiting down there. Youd be surprised though how many insurance plans will give you a provider number with just your medical license and a tax ID, so Id give that a shot for billing your services out of network. The number of plans that would credential you as an actual PCP however would be few if any. That would require a number of criteria such as hospital privileges, 24 hour coverage, etc, etc.
Im very interested though on the kind of services you plan on providing in that basement of yours! You could certainly draw labs yourself, use the pull-out sofa for minor procedures or gyn exams

.. stick an eye chart on the wall. In terms of equipment, as long as you limit yourself to things like med refills and the bare minimum of diagnostics, I guess you dont need much!
Its all a very interesting concept though! I like people who think outside the box!